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Transmission Lines and E.M. Waves
Prof R.K. Shevgaonkar
Department of Electrical Engineering
Indian Institute of Technology Bombay
Lecture-5
Welcome, in the last lecture we introduce the concept of Loss-Less Transmission Line,
we say if the resistance and the conductance per unit length of the Transmission Line is
zero then there are only reactive elements in the Transmission Line so there is no loss of
power because there is no ohmic element in the Transmission Line. So in an ideal
situation the line is lossless if R = 0, G = 0
(Refer Slide Time: 01:10 min)
Then we introduce the concept of the Low-Loss Transmission Line which is more
practical line where the resistive component R is much more less than ωL and G is much
more less than ωC.
(Refer Slide Time: 01:25 min)
If this condition is satisfied then we said that we can create the lines still lossless.
However since the loss is very small as and when we require the calculation of losses
along Transmission Line we can use the value of the attenuation constant α, substituting
this condition that R is much more less than ωL and G is much more less than ωC we
calculated the propagation constant which can be separated into real and imaginary part
and then we got the value of α and β in the approximate form.
(Refer Slide Time: 01:58 min)
So, here this quantity was equal to β and this quantity was represented as α. However one
would notice that the condition which you have defined for the low loss is now in terms
of the primary constants of the line. However in the data sheet for a Transmission Line
the primary constants are rarely mentioned, the parameter which I have mentioned for the
Transmission Line are the effective phase constant on the line or the velocity on the cable
and the attenuation constant of the line either in terms of dBs or in terms of Nepers.
Then one would like to convert this condition R < ωL and G < ωC in terms of this
secondary parameters or in terms of the relationship between β and α. Since these
parameters are available readily in the data sheet if I can establish a condition between
these parameters for low loss nature of the line then I can find out whether a particular
line is low loss at a particular frequency.
So now what we do is starting from this relationship between β and α then we can find
out under what condition we can treat the line as a Low-Loss Transmission Line. Taking
this value of α I can multiply this quantity here by a square of L in the numerator and
square root of L in the denominator similarly I can multiply this quantity by square root
of C in the numerator and square root of C in the denominator.
(Refer Slide Time: 03:40 min)
So now the α can be written in terms of this substitution so I get this value of α which is
equal to
1R
LC . Similarly multiplying by
2L
in the second term we get
can be written as
C in the numerator and the denominator
1G
LC . Multiplying numerator and denominator by ω this
2C
1 R
1 G
 LC 
 LC .
2 L
2 C
(Refer Slide Time: 04:38 min)
And as we have seen earlier this quantity  LC is nothing but β so this I can write as β
1 R 1 G 
in to  

.
 2  L 2 C 
Now from the definition of the low loss
R
G
is much more smaller than 1,
is much
C
L
more smaller than 1 so this whole quantity is much more smaller than 1 so essentially
what we are saying is now α is equal to β multiplied very small quantity or in other words
for low loss the condition now is that α is much more less than β.
Since we know β in terms of wavelengths which is nothing but
2

. Once we know the
wavelength on the Transmission Line I can find out what is the value of β from the data
sheet I can find out what the attenuation constant α is. If it is given in terms of dBs I will
convert that in the Nepers per meter and if this condition is satisfied that the β is much
larger compared to α then the line can be treated as the Low-Loss Transmission Line.
(Refer Slide Time: 06:00 min)
What does this physically mean we know if you travel at a distance of λ on Transmission
Line the phase change is equal to 2π.
Let us say if I travel the distance for a lossy Transmission Line then the amplitude of the
wave will reduce by e–α into the distance traveled which is one wavelength. So if I
consider a wave which travels a distance of one wavelength on Transmission Line then
its amplitude will vary e–αx that is equal to e–αλ.
If I travel a distance of one wavelength on Transmission Line substituting for λ which is
2

from the previous equation so λ is
I get here this is equal to e

2
 .2

.
(Refer Slide Time: 07:20 min)
Since for low loss condition α is much smaller than β this quantity
 .2
is much less

 .2
than 1 that means the amplitude of the wave now reduces to e

and since this quantity
is very small essentially the amplitude reduction in the wave is negligibly small. So in
other words what we are saying is a line can be treated a low loss transmission line
provided the change in the amplitude of a traveling wave is negligibly small over one
wavelength distance. Of course negligibly small is a very subjective number you can
consider one percent as negligible or 0.1 percent as negligible.
Let us say as a reference value we consider one percent is a negligible quantity so when
the amplitude reduces to one percent of its original value then we say that the line can be
treated as a Transmission Line. Since this quantity is very small essentially when this
number becomes approximately one percent that is where the amplitude will reduce by
one percent so if
 .2

is approximately
1
the wave amplitude will reduce by one
100
percent over a distance of one wavelength from here then I can find out what is the
acceptable value of α compared to β.
(Refer Slide Time: 08:54 min)
So in practice for a given line there is nothing absolute whether the line is lossy line or a
low loss line for a given frequency it may be possible that α may be much smaller than β
but when the frequency changes the condition may not be satisfied and the line cannot be
treated as a Low Loss Transmission Line. So for a given loss on Transmission Line
depending upon the frequency the line may be treated like a low loss transmission line or
it may not be treated like a Low Loss Transmission Line.
Let us take a simple example to find out what are the physical parameters which we will
have on Transmission Line if we just take some typical line parameters. So let us say I
have a Transmission Line whose primary constants are given as L = 0.25μH/m, let us say
the capacitance per unit length is 100 pico Farad per meter, let us say the conductance per
unit length is zero for this Transmission Line. So G = 0 and we want to know what
should be the resistance per unit length of the Transmission Line so that the line can be
treated like a low loss transmission line.
Let us say the frequency of operation is equal to 100 mega hertz. From here since I know
the value of L and C and the frequency I can find out what is the value of β. So β is equal
to 2π into frequency which is hundred mega hertz which again is 108 hertz into
L
which is 0.25 x 10-6 or micro henry multiplied by hundred into 10-12 or the pico Farad.
This will be equal to π radians per meter.
(Refer Slide Time: 11:10 min)
Just to take a number if I say α is less than one percent of value of β then I can treat the
line as the low loss transmission line. The α should be approximately
back and substitute this value of α in the expression for α that is α =

100
. Now I can go
1
C
R
2
L
(Refer Slide Time: 11:46 min)
and if I substitute the value of
C

and the value of α which is
as you have taken
100
L
one percent of the value of β then I get the value of R which is less than R = π
approximately 3.14 ohms per meter.
So if I accept alpha value to be one percent or less of the propagation constant β or the
phase constant β then the resistance per unit length of the Transmission Line should be
3.14 ohms per meter at the frequency of hundred mega hertz. Of course as I said the
frequency changes then the acceptable value of resistance will change because the line
may not satisfy this condition at that frequency.
now having understood this as we mentioned earlier until and unless somebody
specifically tells you that the line is a lossy line we take a liberty to create the line as a
loss less line because as first order as we have seen that the phase constant for a low loss
line and a lossy line is same, also the characteristic impedance of a low loss line is almost
real and that is same as the characteristic impedance of loss less line.
So here onwards until and unless somebody specifically say include the losses in the
calculation of Transmission Line we will treat the line to be lossless and carry out all our
analysis for a Loss-Less Transmission Line. So essentially we will assume that
characteristic impedance of Transmission Line is given by this so Z0 =
C
. This
L
quantity is a real number and also the propagation constant is equal to the phase constant.
(Refer Slide Time: 14:00 min)
So we have γ = jβ that is again equal to jω
C
. Now with these parameters we will again
L
revisit the voltage and current expressions on Transmission Line and then we carry out
the analysis of the standing waves on the Transmission Lines.
Going back to the original equation of voltage and current as we have seen for a
Transmission Line whose origin has been defined at the load point as L = 0
(Refer Slide Time: 14:30 min)
the voltage and current equation can be given by this. This represent the forward
traveling wave, this represent the backward traveling wave and now we will replace this
quantity γ by jβ where x will be replaced by -1.
So now we have the voltage and current as a function of distance on the Transmission
Line and γ will be replaced by jβ and Z0 is a real quantity. So we can write explicitly the
voltage and current on a Loss-Less Transmission Line. So the voltage is v as a function
of l that is equal to V+ ejβl + V- e-jβl, by taking V+ ejβl common the same thing can be
written as V+ ejβl {1 +
V - -j2βl
e }.
V
And this quantity as we already know is nothing but the reflection coefficient at the load
Vend so this quantity we denote by the reflection coefficient at the load end ΓL so  as
V
we have seen earlier is nothing but equal to ΓL which is equal to
Z L  Z0
.
ZL +Z0
So now the voltage at any location on the Transmission Line can be given as V+ ejβl {1 +
ΓL e-j2βl}.
(Refer Slide Time: 16:55 min)
Similarly I can take the current equation I can substitute γ = jβl in the current equation I
V  j l V  -j l
e 
e
can get the current that any location on the line I(l) =
Z0
Z0
V  j l
V  j l
e common we can write down here this is
e {1 - ΓL e-j2βl} where
Again taking
Z0
Z0
Vwill be equal to ΓL.
V
(Refer Slide Time: 18:08 min)
These two terms as we know essentially represent the forward and backward traveling
wave so the whole expression here essentially represent super position of the forward and
the backward traveling wave which is nothing but a standing wave on a Transmission
Line.
So here the expression for voltage and current represent the standing voltage and standing
current wave on the Transmission Line.
Now we can investigate certain features for the Transmission Line from here and first
thing what we will note is that there are two terms either in voltage or current so you are
having this term which is having amplitude one and then you arriving at second term
whose amplitude is modulus of this quantity ΓL plus a phase which is the phase of ΓL plus
this quantity phase which is minus j2βl. Writing very explicitly the complex reflection
coefficient in terms of its magnitude and phase let us say I define ΓL = |ΓL| e jL where L
is the phase of the reflection coefficient at the load end.
(Refer Slide Time: 19:41 min)
Then I can write down the current and voltage explicitly in terms of this magnitude of
reflection coefficient and the phase. So finally I have two expressions here one for
voltage V(l) = V+ ejβl {1 + ΓL e j -2 l } and the current I(l) will be
(Refer Slide Time: 20:40 min)
V  j l
e {1 - ΓL e j -2 l }.
Z0
Now let us see how the voltage and current varies if I measure the magnitude of the
voltage and current along the Transmission Line, what is the variation of the voltage and
current along the Transmission Line.
So first thing what we will notice is as you move along the Transmission Line this
quantity L is positive because we are moving towards the generator so as I move towards
the generator the phase becomes more and more negative so this quantity ( L - 2βl) phase
becomes more and more negative or in terms of a complex plane when the phase become
more negative essentially we will move on the clockwise direction. So by moving
towards the generator the phase becomes more and more negative the amplitude of this
thing remains constant this term and the total voltage will be the vector sum of this term
and this term is the real term, this term is the complex term whose phase is given by that
and whose magnitude is given by |ΓL|.
So essentially this is saying that this is the vector of unity one which represent the first
term then I have another vector whose magnitude is |ΓL| and the phase of this is ( L - 2βl)
and as I move towards the generator the phase becomes more negative but the magnitude
of this remains constant. That means this whole quantity represent the circle where the
point moves on this circle as the L changes. So this motion around this is towards
generator and the magnitude of this term in the curly brackets is given by the vector
which is the joining of these two points so this is nothing but |{1 + ΓL e j -2 l }| magnitude
of this quantity.
(Refer Slide Time: 23:03 min)
The first term is this vector which is unity the second term is this vector which is rotating
as we move towards the generator on the Transmission Line and the magnitude of the
total quantity here essentially varies as we move on the Transmission Line. Now the
thing to note is when it moves on a circle at some phase when this quantity is zero or 2π
or 4π ej0 or ej2π or ej4π that is equal to +1.
So I get a magnitude which is maximum which is represented by this point that is nothing
but 1 + |ΓL|. Similarly if this quantity ( L - 2βl) is odd multiples of  ej odd multiples of π
will be equal to -1 so I will get one minus mod |ΓL| that is the minimum value which I
will see for this term here which is represented by this point where the two terms cancel
each other.
I see that the variation of voltage and current is bound by two limits when the two terms
directly add each other that time I will see the maximum voltage. If they cancel each
other i will see the minimum of voltage similarly when these two terms add each other I
will get the maximum current and when these two terms cancel each other I will get
minimum of the current.
So the condition is when this quantity is +1 the voltage will become maximum but when
this quantity is +1 and this quantity is +1 there is a minus sign here so when this quantity
goes maximum at the same location l this quantity will go minimum that’s a very
interesting thing now. Earlier when we talked about lumped circuit wherever we have
voltage higher we also have the current higher. Now what we are seeing here is that when
the voltage is maximum the current is minimum and vice versa when this quantity
become -1 that time the voltage will be minimum but this quantity will become plus so I
will get the current maximum. Or in other words if I measure the magnitude of the
current and voltages on the Transmission Line the maximum current and maximum
voltages do not occur at the same location rather they are staggered in space. Wherever
there is maximum voltage there is minimum current and vice versa. So the standing wave
of the voltage and current are shifted with respect to each other in space on the
Transmission Line.
So if I plot the magnitude with the voltage on the Transmission Line so let us say this is
my Transmission Line which is terminated in some load here this is ZL and now I plot the
voltage and current on the Transmission Line let us say I plot |V| the |V| will have a
variation which will go something like that this is the location where magnitude of
voltage is minimum, this is the location where the magnitude of the voltage is maximum
and as we saw just now that wherever the voltage is maximum the current will be
minimum and vice versa so the current will go something like that so this is the plot for
voltage |V| and this is plot for modulus of current. So this is this plot we are having |V| or
|I| as a function of distance l = 0 and distance is measured towards the generator.
(Refer Slide Time: 27:30 min)
So in this location I get voltage maximum and current minimum, when I go here I get
current maximum and voltage minimum. The important thing is when the standing waves
are present on the Transmission Line then the voltage and current waveforms are shifted
with respect to each other and maximum of voltage aligns with minimum of current and
vice versa.
Now having understood this I can again go back and look at this expression of the voltage
and current. If I go to the location where the voltage is maximum that means this quantity
is +1 the voltage will be |V+| multiplied by 1 + ΓL because this quantity is +1 at the same
location I will have the current which will be minimum as we saw so this will be I(l) =
V
Z0
(1 - ΓL) where Z0 is real since the line is loss less.
So once I know the voltage and current I can find out the impedance at this location
where the voltage is maximum or the voltage is minimum or when the current is
minimum or the current is maximum. The interesting thing to note here is irrespective of
phase of V+ when the voltage is maximum or minimum the ratio of V and I is a real
quantity. If I take a ratio of these two
1   L e j(L  2  l ) 

V  l

that is equal to Z0 
and
j(L  2  l ) 
1
e
I l


L


when voltage is maximum or minimum this quantity is +1 or -1 so for maximum voltage
( L - 2βl) is even multiple of π that is it is 0, 2π, 4π and so on, for minimum voltage ( L 2βl) is equal to odd multiple of π that is π, 3π, 5π and so on.
(Refer Slide Time: 30:21 min)
So when the voltage is maximum this quantity is +1 so Z0 is real, now this quantity is one
|1 + ΓL| denominator this quantity is again +1 so this quantity is |1 - ΓL| so the ratio of this
quantity is the real quantity. When the voltage is maximum the impedance seen on the
Transmission Line is real irrespective of what the line is terminated in.
Even if the line is terminated in complex impedance if I move on the Transmission Line
and go to the location where the voltage magnitude is maximum, at that location the
impedance measured will be always real. Similarly when I go to a location where the
voltage is minimum this quantity will be equal to -1so I will get |1 - ΓL| you have |1 + ΓL|
now. And again this quantity will be real so now we make a very important conclusion
and that is on a Transmission Line wherever there is a voltage maximum the impedance
measured is real, wherever there is a voltage minimum the impedance measured is real.
So irrespective of what the impedance with which the line is terminated you will always
find these points on the line where the voltage is maximum or minimum and that location
the impedance measured will be purely a real quantity.
What will the value of these maximum or minimum impedances? We can substitute this
so at a location where the voltage is maximum and the current is minimum that is the
highest impedance you are going to measure on the Transmission Line. So this quantity
when the voltage is maximum at that location we get the maximum possible impedance
which we can measure on the Transmission Line. So we can get the maximum impedance
which one can see on the Transmission Line and let us call that as Zmax is nothing but
|Vmax |
.
|I min |
(Refer Slide Time: 32:52 min)
And since we have seen that this quantity when you are having the voltage maximum and
current minimum that time the phase difference between them is zero so this quantity is
real quantity so Zmax is nothing but Rmax as resistive impedance, from here if I substitute
1   L
this equal to plus one I will get Rmax = Z0 
 1-  L

.

Similarly if I go to a location where the voltage is minimum so this value will be |1 - ΓL|
but at the same time the current will be maximum so that is the lowest value of
impedance you can see on the transmission line. So you get the minimum impedance on
the line which is Zmin which will be

 1-  L
|Vmin |
and that will be equal to Z0 
|I max |

1+  L


.


(Refer Slide Time: 34:14 min)
So once the load impedance on the line is known the reflection coefficient ΓL is known its
magnitude is known. I know what the maximum and minimum value of impedance is I
can see on the Transmission Line. So as we move on the Transmission Line the
impedance is going to vary as we saw because of impedance transformation but there is a
bound on this impedance variation the lowest value of impedance which one can see on
Transmission Line is Zmin or Rmin is resistive and the maximum value which one can see
on the Transmission Line is Rmax which is given by that.
Now once we are having the voltage standing wave on Transmission Line at high
frequencies the measurement of phase is rather complicated. You can measure the
amplitude of the signal rather reliably but the measurement of phase is little uncertain. So
at high frequency normally we have an effort to estimate the phase not in the direct
manner but in indirect manner. By carrying out the measurements of only magnitude kind
of quantities we would like to estimate the phase of the signal and as we have seen that
the phase of the signal in time get translated into the phase space because the total phase
which we seen on a wave is a combination of space and time or in other words the phase
relationship between the two waves the forward and the backward traveling wave that is
related to the time phase as well as the space phase.
And since this total phase governs the location of maxima and minima of the standing
wave noting the location of maximum and minimum on Transmission Line one can
estimate the phase which is there with the signal. So what we do now is we define a
parameter for the standing wave which is a parameter of only amplitude variation on the
Transmission Line and that quantity is called the voltage standing wave ratio. It is
essentially a measure of what is the relative contribution of the reflected wave to the
incident wave if the reflected wave is zero then there is no standing wave you will have
only traveling wave if the reflected wave is full then you will have completely developed
standing wave.
So the interference of the two waves the forward and the backward waves are going to
give me this variation of the maximum to minimum. So we define this quantity the value
which you get for the maximum magnitude on the standing wave and the minimum
amplitude on the standing wave if I call the maximum value as Vmax and the minimum
value as Vmin then the ratio of Vmax to Vmin magnitude is the voltage standing wave ratio.
And this quantity is a very important quantity because without carrying out any phase
measurement we can measure this quantity on the Transmission Line. Recall reflection
coefficient is a complex quantity so if you want to have the complete knowledge of the
reflection coefficient then we have to get its amplitude and phase. However the quantity
which you are defining now is called the voltage standing wave ratio which is measured
by only amplitude measurement.
So by measuring the maximum and minimum magnitude of the standing wave we get this
quantity called the voltage standing wave ratio, normally it is denoted by ρ =
V max
V min
.
(Refer Slide Time: 38:47 min)
And what is the maximum value of voltage which I can see on the line is |V |1+ L 
and the minimum value which I can see on the Transmission Line is |V  |1- L  .

1+  L
The |V+| cancels so the voltage standing wave ratio is 

 1-  L


 , in short the voltage


standing wave ratio is called as VSWR. So the VSWR for a load whose reflection
coefficient magnitude is  L is given by this.
(Refer Slide Time: 39:50 min)
Since the line is lossless the reflection coefficient at the load is
ZL  Z 0
and Z0 is the real
ZL  Z 0
quantity for a Loss-Less Transmission Line so ZL can have any complex impedance
terminated on the line but Z0 is real without much effort one can see that the magnitude
of this quantity is all ways less than one. So the mod of  L is always less than or equal
to 1 and that is makes a physical sense because what
 L is telling you is the relative
amplitude of the reflected wave compared to the incident wave.
(Refer Slide Time: 40:42 min)
Since we do not have any energy source on the load point the part of the energy only can
get reflected so the amplitude of the reflected wave has to be always less than or equal to
the incident wave. So for any passive loads the  L is always less than or equal to 1 so in
a condition when ZL = 0 or ZL = ∞ I will get the magnitude of  L = 1 otherwise this
quantity will be always less than or equal to one so if I want to see very specific load
impedances for which the  L condition will be satisfied. I will see there will be three
cases. Case one will be when ZL = 0 that means the line is short circuited at the load end
so this is a condition for a short circuited line. I substitute ZL = 0 so I get  L = -1 in this
case I get  L = -1 or  L = 1.
(Refer Slide Time: 42:10 min)
The second case if I take ZL = ∞ that means the line is open circuited then again I can
substitute ZL = ∞, take ZL first common so it will become
1 – Z0
so this will be equal to

+1. So this will give me  L = +1 giving me again  L = +1.
The third case is that if the line is terminated in an ideal reactance so the third case is if
ZL = j times x your reactance then again so this is pure reactance then  L will be equal to
jx  Z0
.
jx + Z0
This quantity in general will be complex but the  L will be equal to again +1.
(Refer Slide Time: 43:46 min)
So we have these three situations when the line is short circuited, when the line is open
circuited at the load end or when the line is terminated in a pure reactance the mod of
reflection coefficient is one. What that means is if the line is either short circuited or open
circuited or terminated in an ideal reactance there will be full reflection from the load end
and that make sense that since I am having short circuit or open circuit or pure reactance
there is no energy absorbing element available at the load end of the line neither short
circuit can absorb power nor open circuit can absorb power nor a ideal reactance can
absorb power.
So whatever power the traveling wave takes to the load end there is no option but to take
the entire power back in the reflected waveform. Whatever wave reaches to the load end
that completely get reflected if any of these three conditions are satisfied and that is what
essentially is represented by this that the magnitude of reflection coefficient becomes
equal to one. That means the entire energy with which reaches to the load it gets reflected
so what we see from here is the reflection coefficient is always less than one it becomes
equal to one in this special cases that is short circuit, open circuit or ideal reactance other
wise the magnitude of the reflection coefficient is always less than one.
Now if I substitute this into the quantity which we have defined voltage standing wave
ratio now this quantity  L is less than or equal to one. So in the best case when this
quantity is zero we will get a VSWR equal to one otherwise this quantity will be always
greater than one. So as we got the upper bound on reflection coefficient  L . So we can
also define the bounds on the voltage standing wave ratio VSWR and that is when  L =
0 then the VSWR = 1. However when  L goes to 1 that time this quantity is infinity so
we have VSWR ρ its bounds are one equal to or less than equal to infinity.
(Refer Slide Time: 46:22)
And which is the better case? The  L = 0 represents no reflected wave on Transmission
Line that means we have full power transfer to the load. So  L going to zero is the best
case as far the reflection is concerned or in other words when VSWR = 1 that is the best
case for the power transfer on the line. As the VSWR increases it indicates higher and
higher value of  L that is higher and higher reflection on the Transmission Line or
lesser and lesser efficiency of power transfer to the load.
So in every circuit design our effort is to make the VSWR as small as possible or as close
as to 1 as possible. Higher the value of VSWR indicates more mismatch on the
Transmission Line or higher value of the reflected wave on the Transmission Line.
So this quantity VSWR is one of the very important quantity in the measurement at high
frequencies. Whenever we design a circuit we essentially try to measure the VSWR on
the Transmission Line and we try to make the VSWR as close to one as possible, making
sure that the circuit is efficiently transferring power to the load end of the line.
Now once you have defined this parameter ρ then one can relate this ρ to the maximum
and minimum impedance which one can see on Transmission Line. We have already seen
that the maximum value of the impedance which we can see on the line is Rmax = Z0

1+  L


 1-  L


 but now you know this quantity is a measurable quantity and that is nothing


but the voltage standing wave ratio ρ.
So the maximum value of the resistance or the impedance which we can again see on the
line is nothing but Z0 into wave standing wave ratio ρ.
(Refer Slide Time: 48:34 min)
Similarly the minimum resistance which I will see on the line is this is
to
Z0

1

so this is equal
.
So I get from here Rmax = Z0 ρand Rmin on the line is equal to
Z0

. So for any line which
is terminated in arbitrary impedance if I move to a location where the voltage is
maximum then I know the value of the impedance at that location because measuring the
voltage maxima and minima I can find out the value of this voltage standing wave ratio ρ,
I know the characteristic impedance align apriori so I know the maximum value of the
impedance which line will show.
(Refer Slide Time: 49:33 min)
Similarly at the location where the voltage is minimum the impedance measured will be
Rmin and that will be the characteristic impedance divide by the voltage standing wave
ratio. Now if you recall we had mentioned that if the impedance is known on any point on
Transmission Line then you can always transform that impedance to any other location
on the Transmission Line. Then immediately it strikes to us that knowing the location at
the voltage maximum or minimum the impedance value is known there. so now I can
transform either side of Transmission Line towards the load so that I can get the load
impedance.
Now this essentially opens a measurement technique for the unknown impedance at high
frequency if you are having a complex impedance its measurement is quite tedious
because we cannot re-measure the phase very accurately. But I have a mechanism of
measuring the phase indirectly as we already mention the phase get reflected into the
standing wave pattern or the location of the voltage maxima and minima. So if I measure
the voltage standing wave ratio and the location of the voltage maxima and minima I can
always transform this impedance Rmax or Rmin to the location of the load which is nothing
but the load impedance.
So if I transform from the load impedance to voltage maxima I will get this impedances
but I can work backwards and say if I know the location of the voltage maximum from
the load, I know the value of impedance at that location, I know the distance of load from
that point so transformation of this impedance to the load end should give me the load
impedance. When we go to the application of Transmission Lines we will see that this
technique is used for measuring the complex impedances and at that time we will
explicitly derive the expression for the unknown impedance which is terminated to a
Transmission Line.
Thank you.